How to reflect individual bone axis?
I haven't got a full picture of what you want to accomplish with this but:

What you can do is to have an armature that is scaled to -1 on one axis. That will give you an inverted coordinate system inside that. You could then drive bones on the deforming armature with those bones.

AFAIK bones will end up as matrices that deforms the mesh. Having a negatively  scaled matrix on one bone will also scale the deformation accordingly. By using a controlling bone that exist in another coordinate system, this problem will be limited to a non deforming armature, and this is fine.